How to Match a^n b^n c^n Using Regular Expressions (PCRE)?

Match a^n b^n c^n Using Regular Expressions (PCRE)

Regular expression engines have advanced beyond the original theory of regular grammars, enabling them to handle patterns that were previously considered impossible. One such pattern is the context-sensitive grammar {a^n b^n c^n; n>0}, which matches strings containing an equal number of a's, b's, and c's.

This complex pattern can be matched using the following PCRE expression:

~^
    (?=(a(?-1)?b)c)
     a+(b(?-1)?c)
$~x

Explanation:

• The ^ and $ anchors ensure the pattern matches the entire string.
• The positive lookahead assertion (?=(a(?-1)?b)c) checks if, for each occurrence of "ab," there is an equal number of c's.
• The a (b(?-1)?c) group captures an arbitrary number of a's, followed by an equal number of b's and c's.

Key Insights:

• Modern PCRE allows for non-regular patterns to be matched.
• This pattern illustrates the power of lookahead assertions and recursive negations.
• The ability of regex to parse context-sensitive grammars challenges the notion that they are limited to regular grammars.

Example Matches:

• aaabbbccc matches (1)
• aaabbbcc does not match (0)
• aaaccc does not match (0)
• aabcc does not match (0)
• abbcc does not match (0)

This regex demonstrates that PCRE's capabilities extend beyond regular languages, enabling it to process more complex patterns.

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